Judgment and Decision Making in Information Systems Probability, Utility, and Game Theory Yuval...
-
Upload
olivia-warren -
Category
Documents
-
view
219 -
download
1
Transcript of Judgment and Decision Making in Information Systems Probability, Utility, and Game Theory Yuval...
![Page 1: Judgment and Decision Making in Information Systems Probability, Utility, and Game Theory Yuval Shahar, M.D., Ph.D.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d955503460f94a7d8f4/html5/thumbnails/1.jpg)
Judgment and Decision Making in Information Systems
Probability, Utility, and Game Theory
Yuval Shahar, M.D., Ph.D.
![Page 2: Judgment and Decision Making in Information Systems Probability, Utility, and Game Theory Yuval Shahar, M.D., Ph.D.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d955503460f94a7d8f4/html5/thumbnails/2.jpg)
Probability: A Quick Introduction
• Probability of A: P(A)• P is a probability function that assigns a number in
the range [0, 1] to each event in event space• The sum of the probabilities of all the events is 1• Prior (a priori) probability of A, P(A): with no
new information about A or related events (e.g., no patient information)
• Posterior (a posteriori) probability of A: P(A) given certain (usually relevant) information (e.g., laboratory tests)
![Page 3: Judgment and Decision Making in Information Systems Probability, Utility, and Game Theory Yuval Shahar, M.D., Ph.D.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d955503460f94a7d8f4/html5/thumbnails/3.jpg)
Probabilistic Calculus
• If A, B are mutually exclusive:– P(A or B) = P(A) + P(B)
• Thus: P(not(A)) = P(Ac) = 1-P(A)
A B
![Page 4: Judgment and Decision Making in Information Systems Probability, Utility, and Game Theory Yuval Shahar, M.D., Ph.D.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d955503460f94a7d8f4/html5/thumbnails/4.jpg)
Independence• In general:
– P(A & B) = P(A) * P(B|A)
• A, B are independent iff – P(A & B) = P(A) * P(B)
– That is, P(A) = P(A|B)
• If A,B are not mutually exclusive, but are independent:– P(A or B) = 1-P(not(A) & not(B)) = 1-(1-P(A))*(1-P(B))
= P(A)+P(B)-P(A)*P(B) = P(A)+P(B) - P(A & B)
A BA & B
![Page 5: Judgment and Decision Making in Information Systems Probability, Utility, and Game Theory Yuval Shahar, M.D., Ph.D.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d955503460f94a7d8f4/html5/thumbnails/5.jpg)
Conditional Probability
• Conditional probability: P(B|A)
• Independence of A and B: P(B) = P(B|A)
• Conditional independence of B and C, given A: P(B|A) = P(B|A & C) – (e.g., two symptoms, given a specific disease)
![Page 6: Judgment and Decision Making in Information Systems Probability, Utility, and Game Theory Yuval Shahar, M.D., Ph.D.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d955503460f94a7d8f4/html5/thumbnails/6.jpg)
Odds
• Odds (A) = P(A)/(1-P(A))
• P = Odds/(1+Odds)
• Thus, – if P(A) = 1/3 then Odds(A) = 1:2 = 1/2
![Page 7: Judgment and Decision Making in Information Systems Probability, Utility, and Game Theory Yuval Shahar, M.D., Ph.D.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d955503460f94a7d8f4/html5/thumbnails/7.jpg)
Bayes Theorem
TP
DTPDPTDPpositivetestdiseaseP
AP
BAPBPABP
()
(|)()(|)(:|)
()
(|)()(|)
B,(|P(A P(B) A(|P(A)P(B B(&P(A
For example, for diagnostic purposes:
![Page 8: Judgment and Decision Making in Information Systems Probability, Utility, and Game Theory Yuval Shahar, M.D., Ph.D.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d955503460f94a7d8f4/html5/thumbnails/8.jpg)
Expected Value
n
i
ii XPX1
)(E[X]
0
)(E[X] dxxxp
If a random variable X can take on discrete values Xi with probability P(Xi ) then the expected value of X is
If a random variable X is continuous, then the expected value of X is
![Page 9: Judgment and Decision Making in Information Systems Probability, Utility, and Game Theory Yuval Shahar, M.D., Ph.D.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d955503460f94a7d8f4/html5/thumbnails/9.jpg)
Examples
• The expected value of of a throw of a die with values [1..6] is 21/6 = 3.5
• The probability of drawing 2 red balls in succession without replacement from an urn containing 3 red balls and 5 black balls is:– 3/8 * 2/7 = 6/56 = 3/28
![Page 10: Judgment and Decision Making in Information Systems Probability, Utility, and Game Theory Yuval Shahar, M.D., Ph.D.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d955503460f94a7d8f4/html5/thumbnails/10.jpg)
Binomial Distribution
2
4
• The probability of tossing 4 (fair) coins and getting exactly 2 heads and 2 tails:
1/16 * = 1/16 * 6 = 6/16 = 3/8
![Page 11: Judgment and Decision Making in Information Systems Probability, Utility, and Game Theory Yuval Shahar, M.D., Ph.D.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d955503460f94a7d8f4/html5/thumbnails/11.jpg)
A Gender Problem
• My neighbor has 2 children, at least one of which is a boy. What is the probability that the other child is a boy as well? Why?
![Page 12: Judgment and Decision Making in Information Systems Probability, Utility, and Game Theory Yuval Shahar, M.D., Ph.D.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d955503460f94a7d8f4/html5/thumbnails/12.jpg)
The Game Show Problem
• You are on a game show, given the choice of 3 doors. Behind one is a car, behind the 2 others, goats. You get to keep whatever is behind the door you chose. You pick a door at random (say, No. 1) and the host, who knows what is behind the doors, opens another door (say, No. 2), which has a goat behind it. Should you stay with your choice or switch to the 3rd door? Why?
![Page 13: Judgment and Decision Making in Information Systems Probability, Utility, and Game Theory Yuval Shahar, M.D., Ph.D.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d955503460f94a7d8f4/html5/thumbnails/13.jpg)
The Birthday Problem
• Assuming uniform and independent distribution of birthdays, what is the probability that at least two students have the same birthday in a class that has 23 students? Why?
![Page 14: Judgment and Decision Making in Information Systems Probability, Utility, and Game Theory Yuval Shahar, M.D., Ph.D.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d955503460f94a7d8f4/html5/thumbnails/14.jpg)
Lotteries and Normative Axioms• John von Neumann and Oscar Morgenstern
(VNM) in their classic work on game theory (1944, 1947) defined several axioms a rational (normative) decision maker might follow (see Myerson, Chap 1.3) with respect to preference among lotteries
• The VNM axioms state our rules of actional thought more formally with respect to preferring one lottery over another
• A lottery is a probability function from a set of states S of the world into a set X of possible prizes
![Page 15: Judgment and Decision Making in Information Systems Probability, Utility, and Game Theory Yuval Shahar, M.D., Ph.D.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d955503460f94a7d8f4/html5/thumbnails/15.jpg)
Utility Functions
• Assuming a lottery f with a set of states S and a set of prizes X, a utility function is any function u:X x S -> R (that is, into the real numbers)
• One important utility function of an outcome x is the one assessed by asking the decision maker to assign a preference probability among the worst outcome X0 and the best outcome X1
– Note: There must be such a probability, due to the continuity axiom (our equivalence rule)
![Page 16: Judgment and Decision Making in Information Systems Probability, Utility, and Game Theory Yuval Shahar, M.D., Ph.D.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d955503460f94a7d8f4/html5/thumbnails/16.jpg)
The Continuity Axiom
• If there are lotteries La, Lb, Lc; La > Lb > Lc (preference relation), then there is a number 0<p<1 such that the decision maker is indifferent between getting lottery Lb for sure, and receiving a compound lottery with probability p of getting lottery La and probability 1-p of getting lottery Lc
– P is the preference probability of this model– B is the certain equivalent of the La, Lc deal
![Page 17: Judgment and Decision Making in Information Systems Probability, Utility, and Game Theory Yuval Shahar, M.D., Ph.D.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d955503460f94a7d8f4/html5/thumbnails/17.jpg)
Preference Probabilities
1 P
1-P
Lb
B is the Certain Equivalent of the lottery < La, p; Lc, 1-p<
La
Lc
![Page 18: Judgment and Decision Making in Information Systems Probability, Utility, and Game Theory Yuval Shahar, M.D., Ph.D.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d955503460f94a7d8f4/html5/thumbnails/18.jpg)
The Expected-Utility Maximization Theorem
• Theorem: The VNM axioms are jointly satisfied iff there exists a utility function U in the range [0..1] such that lottery f is (weakly) preferred to lottery g iff the expected value of the utility of lottery f is greater or equal to that of lottery g (see Myerson Chap 1)– Note: The proof shows that the preference probability
(and its linear combinations) in fact satisfies the requirements
![Page 19: Judgment and Decision Making in Information Systems Probability, Utility, and Game Theory Yuval Shahar, M.D., Ph.D.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d955503460f94a7d8f4/html5/thumbnails/19.jpg)
Implications of Utility Maximization to Decision Making• Starting from relatively very weak assumptions,
VNM showed that there is always a utility measure that is maximized, given a normative decision maker that follows intuitively highly plausible behavior rules
• Maximization of expected utility could even be viewed as an evolutionary law of maximizing some survival function
• However, in reality (descriptive behavior) people often violate each and every one of the axioms!
![Page 20: Judgment and Decision Making in Information Systems Probability, Utility, and Game Theory Yuval Shahar, M.D., Ph.D.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d955503460f94a7d8f4/html5/thumbnails/20.jpg)
The Allais Paradox (Cancellation)
• What would you prefer:– A: $1M for sure– B: a 10% chance of $2.5M, an 89% chance of
$1M, and a 1 % chance of getting $0 ?
• And which would you like better:– C: an 11% chance of $1M and an 89% of $0– D: a 10% chance of $2.5M and a 90% chance
of $0
![Page 21: Judgment and Decision Making in Information Systems Probability, Utility, and Game Theory Yuval Shahar, M.D., Ph.D.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d955503460f94a7d8f4/html5/thumbnails/21.jpg)
The Allais Paradox, Graphically 10% 89% 1%
$1M $1M $1M
$2.5 $1M $0
$1M $0 $1M
$2.5M $0 $0
A
B
C
D
![Page 22: Judgment and Decision Making in Information Systems Probability, Utility, and Game Theory Yuval Shahar, M.D., Ph.D.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d955503460f94a7d8f4/html5/thumbnails/22.jpg)
The Elsberg Paradox (Cancellation)
• Suppose an urn contains 90 balls; 30 are red, the other 60 an unknown mixture of black and yellow. One ball is drawn.
– Game A: 1. If you bet on Red, you get a $100 for red, $0 otherwise;2. If you bet on black, $100 for black, $0 otherwise
– Game B: 1. If you bet on red or yellow, you get a $100 for either, $0
otherwise; 2. If you bet on black or yellow, you get $100 for either, $0
otherwise
![Page 23: Judgment and Decision Making in Information Systems Probability, Utility, and Game Theory Yuval Shahar, M.D., Ph.D.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d955503460f94a7d8f4/html5/thumbnails/23.jpg)
The Elsberg Paradox, Revisited
30 Balls 60Balls
GameRedBlackYellow
A.1$100$0$0
A.2$0$100$0
B.1$100$0$100
B.2$0$100$100
![Page 24: Judgment and Decision Making in Information Systems Probability, Utility, and Game Theory Yuval Shahar, M.D., Ph.D.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d955503460f94a7d8f4/html5/thumbnails/24.jpg)
An Intransitivity ParadoxDimensions
IQExperience in Years
A1201
ApplicantsB1102
C1003
Decision Rule: Prefer intelligence if IQ gap > 10, else experience
![Page 25: Judgment and Decision Making in Information Systems Probability, Utility, and Game Theory Yuval Shahar, M.D., Ph.D.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d955503460f94a7d8f4/html5/thumbnails/25.jpg)
The Theater Ticket Paradox (Kahneman and Tversky 1982)
• You intend to attend a theater show that costs $50. – A:You bought a ticket for $50, but lost it on the
way to the show. Will you buy another one?– B: You lost $50 on the way to the show. Will
you buy a ticket?
![Page 26: Judgment and Decision Making in Information Systems Probability, Utility, and Game Theory Yuval Shahar, M.D., Ph.D.](https://reader036.fdocuments.in/reader036/viewer/2022062516/56649d955503460f94a7d8f4/html5/thumbnails/26.jpg)
Are People Really Irrational?
• Not necessarily!• The cost of following normative principles,
as opposed to applying simplifying approximations, might be too much on average in the long run
• Remember that the decision maker assumes that the real world is not designed to take advantage of her approximation method